modeling viscoelastic free surface and interfacial flows, with applications to the deformation of droplets and blood cells

The development of viscoelastic free surface flow modeling and isochoric domain deformation method is applied to model cell viscoelastic drop deformation... 5.20 5.21 5.22 Comparison of

RICE UNIVERSITY Modeling Viscoelastic Free Surface and Interfacial Flows,

with Applications to the Deformation of Droplets and Blood

APPROVED, THESIS COMMITTEE:

Kyriacos Zyé ú akis, Co-Chair

Professor of Chémical and Biomolecular Eng

Danny Teg Sorensen

Professor of Computational and Applied Mathematics

Houston, ‘Texas April, 2006

Copyright 2006 by Xie, Xueying

All rights reserved

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Modeling Viscoelastic Free Surface and Interfacial Flows, with Applications to the Deformation of Droplets and Blood Cells

is still in its infancy due to complex physics combined with the numerical difficulties

in three-dimension This thesis extends to three-dimensional flows from the previous studies focused on two-dimensional problem

Modeling viscoelastic free surface flows presents several challenges which include modeling the liquid viscoelasticity, locating free surface boundaries, and implementing

large-scale computations Conformation tensor models are used to model the fluid

viscoelasticity because they balance generality, realistic physics, and computational cost A new, convenient open-flow boundary condition is developed for the transport equation of the conformation tensor The domain deformation method is used to

locate both two- and three-dimensional free surfaces and interfaces by treating the

mesh as an elastic pseudo-solid In addition, an isochoric domain deformation method

is developed to conserve domain volumes for certain free surface flows where the

volume of a liquid domain is prescribed, such as a cell deforming in shear flow

The equations for solving viscoelastic free surface flows are discretized by the finite element method The non-linear discretized equations are solved by Newton’s method and the resulting large set of linear algebraic equations is solved by parallel GMRES preconditioned by a new sparse approximate inverse preconditioner (SPAI)

The parallel solver together with SPAI is scalable in a wide range of capillary and

Weissenberg numbers; tests on benchmark viscoelastic free surface flows show that problems with millions of unknowns can be tackled on Linux clusters

The development of viscoelastic free surface flow modeling and isochoric domain deformation method is applied to model cell (viscoelastic drop) deformation

Acknowledgments

First of all, I would like to thank my advisor Dr Matteo Paspuali for his advice, insight, and constant encouragement Also, I would like to thank my advisor Dr

Kyriacos Zygourakis for his inspirations and input

I would like to thank Dr George J Hirasaki for helping me on fundamental fluid

dynamics I would like to thank Dr Danny C Sorensen and Dr Mark Embree for

providing help on large scale computations I am specially grateful to Dr Zenaida Castillo for working with me in developing a preconditioner to speed the large scale computation

I would like to thank Dr Marek Behr for his help on finite element method and using pre-process and post-process CFD softwares I would like to thank Dr Lawrence Musson for helping me on developing isochoric domain deformation method

I would like to thank all my groupmates for their helpful discussions and their friendship It is their accompany that makes my PhD life colorful, interesting, and enjoyable In particular, Xiruo Wang gives me tremendous support and care all the

time since the day I left my family in China to pursue a PhD degree in Rice university

I would like to thank my parents for their endless love and full understanding of

my absent from them for so many years Also, I owe a lot to my bother, my sister

and their families, for taking good care of my parents and for believing in me Finally, I would like to thank my husband Zhenghong Zhao for his love, care and

support

1.2 Viscoelastic flow and modeling by conformation tensor model with a

1.3 Parallelizable approximate inverse GMRES preconditioner 5

1.5 Isochoric domain deformation method for volume conserved free sur-

1.6 Computing the steady deformation of viscoelastic drops by the finite

Literature Review: Mechanical Models and Computations of Leuko-

3.6.1 Derivatives of elastic pseudo-solid incompressibility equation 74

3.6.2 Derivatives of the domain deformation mesh generation equation 74

3.6.4 Derivatives of the momentum transport equation 75

3.6.5 Derivatives of the velocity gradient interpolation equation 76 3.6.6 Derivatives of the conformation transport equation 77

Modeling Viscoelastic Flow by Developing a New, Convenient Open-

5.2 Conformation and stress boundary conditions for fully-developed, rec-

5.2.1 Standard boundary conditions 4 105 5.2.2 New boundary condition .0.0200004 107

5.3 Specific conformation-tensor models 109

5.5.1 Two-dimensional flow ina planarchannel 111

5.5.2 Two-dimensional flow around acylinder 120 5.5.3 Three-dimensional flow in a pipe and an annulus 127

5.6 New boundary condition in a 4:8:1 three-dimensional contraction flow 138

5.6.1 Comparison of two- and three-dimensional solutions 145

5.7 Conclusions 2 AT 153

6 Parallelizable Approximate Inverse GMRES Preconditioner for Com-

6.2 Mathematical formulation and Jacobian structure 158

Computation of 2-D free surface/interface flows 199 -

7.5.2 A 2-D collapsible membrane channel 203

7.5.38 Capillary rise 2 2 (da 205 Computation of 3-D free surface/interface flows 210

7.6.2 Viscoelastic flow in a 3-D channel with a free surface section 212 7.6.3 Viscoelastic flow in a 3-D channel flow with a free interface section221 7.6.4 Viscoelastic flow in a rod coating fow , 223 Conclusions 0 c c c c Q cv cv v g gà va xa 229

Computing Steady Free Surface Flow with Incompressible Domain

8.6 Conclusions cv v1 v.v và k k sa 246

9 Computing the Steady Deformation of Viscoelastic Drop by the Fi-

Cell diagram A cell is composed of three main parts: nucleus, cyto-

plasm and membrane Each part has its own specific structure

Phospholipid mobility The types of movement possible for phospho- lipid molecules in a lipid bilayer Reproduced from Alberts, Bray,

Lewis, Raff, Roberts and Watson (1994), p480

Three kinds of cytoskeletal filaments From left to right: actin fil- aments, intermediate filaments and microtubule Reproduced from

Alberts et al (1994), p789 0 0.0 00.00.0000 000048

Mechanical properties of filaments Reproduced from Alberts et al

Nucleus structure Reproduced from Alberts et al (1994), p335

Mechanical models of cells developed from a simple homogeneous model

21

25

1.7m The suction pressures for the different curves are 800Pa (1),

1,400Pa (2), 2,100Pa (3) and 2,000Pa (4) Reproduced from Zhelev and Hochmuth (1994) Page 14 000

Interpretation of the eigenvalues and eigenvectors of the conformation

dyadic Reproduced from Pasquali (2000)

Mapping between two domains 2 is physical domain, Qo is reference domain, 2, is computational domain; Tis domain boundary with su-

perscript * denoting fixed boundary and “ denoting free boundary; A

is the edge crossed by two boundary faces 200

4-node and 10-node tetrahedral elements The nodes are numbered

The mapping between physical 10-node tetrahedral element and its reference element The faces and nodes are numbered according to

Profiles of element relationship matrix Mzzg for a 2-D drop in a chan-

nel; the total element is 1,228 Left: original matrix, right: matrix after element reordering by Reverse Cuthill-McKee method NZ is the total number of non-zero entries The band width is reduced from

Profles of Jacobian matrices for a 2-D drop in a channel The matrix dimension is ð, 720 x ð, 720 Top: original matrix, bottom: matrix after reordering by Reverse Cuthill-McKee method NZ is the total number

of non-zero entries The band width is reduced from 5, 252 to 436

Schematic of rectangular channel flow L = 4b = 4a Both inflow and outflow boundaries are fully developed flow with a fixed pressure; all

the other boundaries are statlc solid walls

The tetrahedral mesh of the a = b = 1/4L rectangular channel Ele-

ment size is h = 0.10 in x; and 22 directions and h = 0.2) in x3 direc-

tion, total element number is 14,593, and total element node number

Comvergence rate of the solution computed in the rectangular flow on different meshes The symbols denote the logarithm of the maximum

relative error € on v; versus the logarithm of the dimensionless element

size h The solid line is the best fit In(e) = 2in(h) -0.98

92

93

97

98

5.1

5.2

5.3

5.4

Geometry and boundary conditions for the two-dimensional planar

channel flow The pressure difference drives flow from left to right,

the moving bottom wall drags flow from right to left; thus, both open-

Left: 16 x 16 mesh used for computing the flow in the two-dimensional

planar channel, computed velocity vectors (showing that both bound-

aries have inflow as well as outflow sections) Right: contour lines of

the non-trivial conformation components (M,, = 0 here) Computed with the Oldroyd-B model at We= 5, Ø=0.59

Streamwise conformation component M,, versus local dimensionless

shear rate We(y) = ALyx at the right (z = L) section of the flow in a

planar channel, computed on the 16 x 16 mesh with the Oldroyd-B at

We = 5, 8 = 0.59 The open circles denote the computed values, the

solid line is the analytical solution M,, =1+2We*(y)

Convergence rate of the solution computed in the flow in the planar

channel with the Oldroyd-B model at 6 = 0.59, We = 3 The symbols denote the logarithm of the maximum relative error e on the conforma-

tion component M,; versus the logarithm of the dimensionless element

size h The solid line is the best fit Infe)=1.9In(h) +41

Xiv

111

113

114

5.9

5.6

5.7

5.8

Off-diagonal component of the conformation tensor M;, versus local

dimensionless shear rate We(y) = AL, computed in the planar channel

flow with the Giesekus model (G = 0.59, a = 0.1) The profiles are re-

ported at the z = 0 boundary (top), z = L/2 channel section (middle),

and z = L boundary (bottom) The symbols denote the computational

results, the solid lines are the analytical solution equation 5.13

Difference between the diagonal (normal) components of the conforma-

tion tensor M,, — My, versus local dimensionless shear rate We(y) =

ALyx computed in the planar channel flow with the Giesekus model

(8 = 0.59, a = 0.1) The profiles are reported at the z = 0 boundary

(top), z = L/2 channel section (middle), and z = L boundary (bot- tom) The symbols denote the computational results, the solid lines are the analytical solution equation 5.14

Geometry and boundary conditions for the two-dimensional flow around

a cylinder M BC denotes that different conformation boundary con-

Mesh of the two-dimensional flow around a cylinder Top: The whole

domain mesh with L, = 15, bottom: Magnified mesh of the region 2R<e<2Ro Q Q Q Q Q Q Q nu gà v kg vi kg kia

121

122

inflow boundary conditions in a two-dimensional flow around a cylinder

at We = 0.5 Top: maximum of the scaled error (infinity norm); bottom: sum of the scaled error (1 norm) 126

Geometry and boundary conditions for the three-dimensional flow in

Geometry and boundary conditions for the three-dimensional flow in

a cylindrical annulus The pressure difference drives flow from left to right, the moving inner cylinder drags flow from right to left; thus,

both open-flow boundaries have inflow and outflow sections 128

Unstructured finite element meshes used for computing flow in a cylin-

Convergence rate of the solution computed in the flow in the cylindri- cal pipe with the Oldroyd-B model at 6 = 0.59, We = 1 The symbols

denote the logarithm of the maximum relative error e on the conforma-

tion component M), versus the logarithm of the dimensionless element

size h The solid line is the best fit In(e) = 1.6ln(h)+3.2 133

5.14 Contours of conformation component M;, at the y = 0 plane computed

at @ = 0.59 with the Oldroyd-B (top row), FENE-P (6 = 10, middle

row) and Giesekus (a = 0.1, bottom row) models Three-dimensional

pipe flow, We = 1, left: characteristic mesh size 0.14R, right: char-

acteristic mesh size 0.06R in radial and circumferential directions and

5.15 Unstructured finite element meshes used for computing flow in a cylin- drical annulus Top: Mesh 1, mesh size 0.11R,_ on the inner wall and 0.14R, on the outer wall; bottom: Mesh 2, mesh size 0.12, in axial direction and 0.06R in radial and circumferential direction

5.16 Contours of conformation component M), at the y = 0 plane computed

at 6 = 0.59 with the Oldroyd-B (top row), FENE-P (b = 10, middle row) and Giesekus (œ = 0.1, bottom row) models Three-dimensional annular flow, We = 2.0 Left: Mesh 1; right: Mesh 2 -

reflections about two planes, only one quarter of the flow domain is

considered, and no-penetration, no-shear stress boundary conditions are imposed on the symmetric boundaries A pressure drop together with the fully-developed flow condition is imposed on the momentum equation at the inflow and outflow boundaries, no-slip is imposed at the solid walls The new boundary condition is imposed on the con- formation tensor at the inflow boundary The downstream width of the channel is W,, the upstream width is W, = 4W,, the breadth is

H = 8W,, the length of the upstream section is L; = 12W,, and that

of the downstream section is L, =4W, 10 0-0-0200 ee

Comparison of the computed results at We = 0.90 on the four sides of the downstream plane z = 4W, with two different downstream lengths,

L, = 4W, and L, = 8W, From top to bottom: streamwise velocity u,

Mi, and Ms SS

Unstructured finite element meshes used for computing the flow in a three-dimensional 4:8:1 contraction The top mesh has a sharp corner,

31,905 elements, and 50,417 nodes; the bottom one has a rounded

corner with radius of curvature r = 0.16W,, 32,258 elements, and

XVili

140

142

5.20

5.21

5.22

Comparison of the streamlines, the positive eigenvalues of the rate

of strain D, and the conformation components My, Mj3, and M33

computed in a two-dimensional 4:1 contraction (left) and computed

on the symmetry boundary of the three-dimensional 4:8:1 contraction

(right) The outflow velocity profile at the symmetry boundary of the

three-dimensional contraction has been matched to its two-dimensional counterpart, which yields slightly different values of Weissenberg num-

ber for the two flows (We = 0.49 in 3-D, and We = 0.54 in 2-D) and

thus produces small differences in the computed solution upstream of

the contraction Oldroyd-B liquid, Ø=0.59

Contours of conformation components in the 4:8:1 contraction flow, computed on Mesh 1 (sharp corner) at We = 0.74, 6 = 0.59 From

left to right: Oldroyd-B, FENE-P, and Giesekus; from top to bottom:

My, Mis, and M33 ee ee ee ee

Contours of conformation components in the 4:8:1 contraction flow,

computed on Mesh 2 (rounded corner) at We = 0.90, 6 = 0.59 From left to right: Oldroyd-B, FENE-P, and Giesekus; from top to bottom:

My, M,3, and M33 Note: the contour scales are truncated to highlight

better the distribution of the various components; values above the

scale maxima are displayed in red The maximum values of My; are

48.85 (Oldroyd-B), 31.97 (FENE-P), and 13.86 (Giesekus)

146

148

149

5.23 Contours of the eigenvalues m3 > mz > m, of the conformation tensor,

computed on Mesh 2 at We = 0.90, 6 = 0.59 From left to right:

Oldroyd-B, FENE-P, and Giesekus; from top to bottom: m3, mạ, and m, Note: the contour scales are truncated to highlight better the distribution of the various eigenvalues; values above the scale maxima are displayed in red The maximum values of m3 are 50.25 (Oldroyd-

B), 33.98 (FENE-P), and 14.83 (Giesekus)

Profile of Jacobian matrix arising in 2-D slot coating flow There are

totally 158, 558 non-zero entTi©S ee

Parameter band defines the band width of the Jacobian which will be used for the construction of P band = 3 02.006

Free surface flow in the downstream of the slot coater and boundary

Total CPU time (symbol circle) and CPU time for computing SPAI

Total memory versus band Memory increases linearly and stays low

CPU time versus problem size for SPAIL-GMRES(1) on problem 2

CPU time versus problem size for SPAI-GMRES(16) on problem 2 184

CPU time versus the number of CPUs for SPAI-GMRES(m) on Mesh

10 n = 1,152,702, Krylov — size = 1000, and band = 201 Top:

the CPU time on computing the preconditioner versus the number of

CPU; bottom: the total CPU time versus the number of CPU 186

Memory requirement versus the number of CPUs for SPAI-GMRES(m)

on Mesh 10 n = 1,152, 702, Krylov — size = 1,000, and band = 201

Top: the memory requirement for storing the preconditioner versus

the number of CPU; bottom: the total memory requirement versus

Convergence history of GMRES when solving 3-D rod coating flow on

Mesh 5 Top: using the first stopping criterion (10~!°); Bottom: using

the second stopping criterion (changable) 4 189 Schematic of an interface between two liquids 195 The illustration of the node duplication on the interface: original node index (left) and new node index after interface node duplication (right) 197 Flow in the downstream of the slot coater: free surface model (top)

and interfacial model (bottom) 6.6 eee ee ee 199

Meshes computed as part of solution of the flows in the slot coater

Q =0.5, Re=0, Ca—Ú.1 ca 200

Computed free surface/interface by three different flows in the slot

Pressure profiles on the bottom line in three different flows at Re = 0 and Ca= 0.1 ’o’ and ’A’ are reference flow 202

Geometry of the 2-D collapsible channel; the segment DC is an elastic

Deformed wall shape with steady flow in the 2-D collapsible channel,

obtained at Re = 1, p} = 9.3 x 104, and y* = 73/@ The solid line

denotes the result of quadrilateral mesh in this study, the symbol ’x’ denotes the result of triangular mesh in this study, and the symbol ’o’

is the result of reference (Luo and Pedley 1995) 04 204

The computed mesh of the 2-D collapsible channel, obtained at Re = 1,

Schematic of the rotating bucket flow 00 210

Computed meshes (left) and velocity vector (right) in the rotating

bucket flow at different Froude numbers From top to bottom: Fr =

Computed and analytical free surface shape at z3 = 0 plane for the

rotating bucket flow at different Froude numbers The symbols denote the finite element solutions and the solid lines are the analytical solutions.212 3-D cylindrical channel with a free surface section and boundary con-

Mesh convergence in the 3-D cylindrical channel with a collapsible

section Ca = 0.122 and We=0.891 2 2 ee ee ee ee 214 The computed mesh in the 3-D cylindrical channel with a collapsible section Ca = 0.122 and We = 0.687 2 2 ee 214 Comparison of the free surface shapes of Newtonian flow and vis-

coelactic flow in a 3-D cylindrical channel with a collapsible section

The effect of We on free surface shape in the 3-D cylindrical channel

with a collapsible section at Ca =0.122 Contours of the conformation tensor in a 3-D cylindrical channel with

a collapsible section on Mesh 3 at Ca = 0.122 and We=0.318 Contours of the conformation tensor in a 3-D cylindrical channel with

a collapsible section on Mesh 3 at Ca = 0.122 and We=0.318

Double layered 3-D collapsible cylindrical channel and boundary con-

ditions Ry = 1, tà = 1.5, Ly = Lg = 1, ha =0.5 048

Comparison of the computed free surface and free interface of Oldroyd-

B flow on the y = 0 plane in the 3-D collapsible channel 3-D rod coating flow and boundary conditions R,; = 1, Re = 2Ri,

Ly = 2h, Tạ = 6A} Be ee ee ee ee

The outer boundary shape of a slice of the 3-D rod coating flow com- puted on Mesh 1 and Mesh 2 at Ca=landWe=1

The computed mesh and stream-wise velocity contours in the 3-D rod

coating flow at Ca= land We=1 0.0.0.0 04

The effect of Ca on the free surface shape in the 3-D rod coating flow

XXiv

219

Axial curvature R, and radial curvature R, on free surface in a 3-D rod.226

The effect of We on the free surface shape in the 3-D rod coating flow

Total normal stress on the free surface in the 3-D rod coating flow at

Contours of conformation components in the 3-D rod coating flow,

computed on Mesh 3 at Ca = 1 and We=0.9

Polymer molecular shapes along free surface in the 3-D rod coating

flow at Ca = 1 and We = 0.9 The color values are proportional to the largest eigenvalues of conformation tensor

Schematic of a constant volume of liquid between two parallel plates with a meniscus The numbers 1 — 4 denote the boundaries, and the letters œ — đ denote the Intersecftion cOrners Reference and deformed physical meshes of the capillary tube (2D) with

a conserved volume of fluid Left: reference mesh, right: computed

Free surface shape of the capillary tube (2D) with a conserved volume

Contours of the liquid pressure p and mapping pseudo-pressure 7 in the

two parallel plates (2D) with a conserved volume of fluid Computed

bers The volume of the domain is conserved Left: Fr = 0.1; right:

FPr=O.4 000 0 ee 245 Velocity vectors for a rotation bucket flow at different Fround numbers The volume of the domain is conserved Left: Fr = 0.1; right: Fr = 0.4 245

Computed and analytical free surface shape at x3 = 0 plane for a rotat- ing bucket flow at different Froude numbers The symbols denote the

finite element solutions and the solid lines are the analytical solutions

Schematic of drop shape and orientation L is the maximum radius,

B is the minimum radius, and @ is the orientation angle between the maximum radius direction and the z, axis 2 2 eee 249

Schematic of a periodic suspension of 2-D drops in a channel flow The domain of the computation is the middle domain with length L 253

Unstructured element node pairs on the left and the right boundaries

Meshes for 2-D periodic drop deformation flow The element size on

the drop surface is refined from (a) to (c) 2.1 eee ee eee 256

Mesh convergence on deformation parameter D of a 2-D periodic New-

tonian drop deforming in a Newtonian matrix at different Ca 257

Comparison of the 2-D drop surface curve in polar coordinates on three

meshes for a 2D periodic Newtonian drop deforming in a Newtonian

matrix at Ca = 0.4 r and a are the drop surface loations in polar

coordinates with the coordinates origin in the drop center

Mesh convergence of the drop orientation angle @ for a 2-D periodic

Newtonian drop deforming in a Newtonian matrix at different Ca

D versus Ca for a 2-D periodic Newtonian drop deforming in a New- tonian matrix; comparison of this work with Zhou & Pozrikidis (1993)

6 versus Ca for a 2-D periodic Newtonian drop deforming in a Newto-

nian matrix; comparison of this work with Zhou & Pozrikidis (1993)

Pressure contours of a 2-D periodic Newtonian drop deforming in a

Velocity contours of a periodic Newtonian drop deforming in a Newto- nian matrix at Ca = 0.4 Top: stream-wise velocity contours, bottom:

Mapping pseudo-pressure 7 contours of a periodic Newtonian drop deforming in a Newtonian matrix at Ca=0.4 -.-

Streamlines of a periodic Newtonian drop deforming in a Newtonian

Comparison of the drop deformation between a single drop and periodic

in an Oldroyd-B fluid The element size on the drop surface is refined

D versus We for a Newtonian drop deforming in an Oldroyd-B matrix

at Ca = 0.1 Mesh convergence on three meshes and comparison with

D versus We for a Newtonian drop deforming in an Oldroyd-B matrix

at Ca = 0.2 Mesh convergence on three meshes and comparison with

Contours of conformation tensor components for a Newtonian drop deforming in an Oldroyd-B matrix at Ca=0.landWe=1 Shape and orientation of ensembles of polymer molecules along drop surface for a Newtonian drop deforming in an Oldroyd-B matrix at

Ca = 0.1 and different Weissenberg number 0-, Orientation angle of polymer molecules, tangent vector angle, and drop radii versus drop surface location angle in polar coordinates for a New-

tonian drop deforming in an Oldroyd-B matrix at Ca = 0.1 and differ- ent Weissenberg number - Q Q Q Q h h hn

Normal stresses along the outer layer of the drop surface for a Newto- nian drop deforming in an Oldroyd-B matrix and comparison with the

D versus We for an Oldroyd-B drop deforming in a Newtonian matrix

at Ca = 0.1 and comparison with Yue et al (2005)

D versus We for an Oldroyd-B drop deforming in a Newtonian matrix

at Ca = 0.2 and comparison with Yue et al (2005)

Contours of the conformation tensor components for an Oldroyd-B drop deforming in a Newtonian matrix at We=1l and Ca=0.1 Orientation angle of polymer molecules, tangent vector angle, and drop

radii versus drop surface location angle in polar coordinates for an

Oldroyd-B drop deforming in a Newtonian matrix at Ca = 0.1 and

Normal stresses along the inner layer of the drop surface for an Oldroyd-

B drop deforming in a Newtonian matrix and comparison with the N/N

D versus We for an Oldroyd-B drop deforming in an Oldroyd-B matrix

at Ca = 0.1 and comparison with the results of other cases including

Rate of change of conformation due to internal processes for several

models of polymer dynamics: Maxwell/Oldroyd-B (Larson 1988); John-

son and Segalman (1977); Larson (1984); Leonov (1976); Giesekus (1982); FENE-P (Bird, Curtiss, Armstrong and Hassager 1987); FENE-

CR (Chilcott and Rallison 1988) Ij and Im are the first (trace) and

second invariant of M Reproduced from Pasquali (2000) with permis-

Quadrature for unit tetrahedra; number of points = 5 and degree of

precision = 3 Cited from Akin (1998) 2 2 eee ee,

Quadrature for faces on unit tetrahedra; number of points = 3 and

degree of precision = 2 Cited from Akin (1998)

Quadrature for unit triangles; number of points = 3 and degree of

precision = 2 Cited from Akin (1998)

Quadrature for edges on unit triangles; number of points = 3 and degree of precision = 2 Adapted from Akin (1998)

Computational error of velocity component v1 computed on a sequence

of increasingly mesh sizes in the rectangular channel The relative error

is calculated by dividing the maximum absolute error by the maximum value obtained with the analytical solution

implementation Notes: 1 Method yields larger bandwidth in linear

system; 2 Method yields degraded accuracy near boundary; 3 Enforcing

the boundary condition is always cumbersome

Computational error on the components of velocity and conformation computed on a 16 x 16 mesh at We = 1 — —9.28, 6 = 0.59 in the

2-D channel The new conformation boundary condition (M6) is im-

posed at the inflow regions The relative error is calculated by dividing the maximum absolute error by the corresponding maximum value ob-

Computational error on the components of velocity and conformation computed on a 16 x 16 mesh at We = 3 and 3.77, 6 = 0.59 in the

2-D channel No boundary condition is imposed on the conformation tensor The relative error is calculated by dividing the maximum ab- solute error by the corresponding maximum value obtained with the

Computational error on the components of conformation computed on

a sequence of increasingly refined meshes with the Oldroyd-B model

at We = 3.0, @ = 0.59 in the 2-D channel The relative error is cal- culated by dividing the maximum absolute error by the corresponding

maximum value obtained with the analytical solution

106

116

117

in axisymmetric pipe flow of an Oldroyd-B liquid at We = 1 to 4,

8 = 0.59; computed as three-dimensional flow on an unstructured tetrahedral mesh with average size 0.25R The relative error is cal- culated by dividing the maximum absolute error in each component

by the corresponding maximum analytical value of that component 131 Computational error on the components of the conformation tensor in

axisymmetric pipe flow of an Oldroyd-B liquid at We = 1, Ø = 0.59;

computed as three-dimensional flow on seven increasingly refined un- structured tetrahedral meshes The relative error is calculated by di- viding the maximum absolute error in each component by the corre- sponding maximum analytical value of that component 132 Computational error on the components of the conformation tensor in axisymmetric annular flow of an Oldroyd-B liquid at We = 0.7 to 2,

8 = 0.59; computed as three-dimensional flow on unstructured tetra-

hedral meshes (figure5.15) The relative error is calculated by dividing

the maximum absolute error in each component by the corresponding

maximum analytical value of that component - 137 Structure of Jacobian matrix ./: related by problem equations, V:

Definition of nine meshes used in the tests Mesh 1 is for the 2-D flow and the rest are for the 3-D flow 0 0.2 0.02004 174

Comparison of the performance of different solvers by solving slot coat-

Comparison of the performance of different solvers by solving 3-D rod coating flow at We = 1 and Ca = 1 FS performance degrades ILUT- GMRES doesn’t converge (DC) SPAIL-GMRES(m) works well

The effect of band on the performance of SPAI-GMRES When band =

41, GMRES doesn’t converge Optimal value: band=61 Parallelization results on Mesh 10 n = 1,152,702, Krylov — size = 1,000, and band = 201 Note: since there’s no results obtained on one

CPU, the parallel speed-up and efficiency are based on 16 CPU results,

ie., D6 = 16 x TTịs/TT,, and PS, = 16x PT¿/PT, 185

Chapter 1 Synopsis

This thesis models 3-D viscoelastic free surface flows by the finite element method, and provides research work on cell deformation in shear flows The computation of cell deformation is motivated by the fact that the astronauts have depressed immune system functions during or after space flight (Taylor and Janney 1992) Research show that the immune cell lymphocyte has decreased number and depressed functionality due to the microgravity environment (Cogoli, Bechler, Muller and Hunzinger 1988,

Cogoli 1993, Taylor and Dardano 1983, Taylor, Neale and Dardano 1986) Variation

of gravity not only changes the body force on lymphocytes, but also alters fluid flow

in the entire human body (Cogoli 1993, Woodman 1995, Stout, Watenpaugh, Breit and Hargens 1995) These changes affect lymphocyte morphology Apart from space flight application, the study of cell deformation is also important for understanding cell culture and blood cell circulation in human body Predictions of shear stress

on cells and analysis of cell deformation under such stresses are important both for designing a bioreactor and understanding cell circulation through capillary vessels when studying disease recovery and wound healing

~ Previous studies have shown the complexity of modeling a cell Cell has instant elastic response to imposed external force; an incompressible solid model that lumped all parts of the cell into one homogeneous material was first suggested (Bagge 1976, Schmid-Schonbein, Sung, Tozeren, Skalak and Chien 1981) However at large de-

formation, cell has fluid-like behavior and thus a Newtonian liquid drop model is

developed (Evans and Kukan 1984) Newtonian assumption was then modified to be

viscoelastic (Dong, Skalak, Sung, Schmid-Schénbein and Chien 1988) to reflect the instant deformation response to external force

In this study, a viscoelastic drop model is selected to compute the cell shape

in shear flow system In this model, the cell internal fluid is a homogeneous vis- coelastic fluid, the external suspending fluid can be either a Newtonian liquid or a non-Newtonian liquid depending on cell environment, and the membrane is a free interface Therefore, modeling cell deformation presents challenges like modeling vis- coelastic fluid flows, modeling free surface/interface flows, and solving large scale equations resulting from computing viscoelastic free surface flows by finite element

method

1.2 Viscoelastic flow and modeling by conformation tensor

model with a new developed boundary condition

Viscoelastic fluid flows arise in disparate processes in engineering, science, and biology—for example, in polymer processing, coating, emulsions, microfluidics hemo- dynamics and many others Since viscoelastic fluids have microstructures that con- tribute to the elasticity of the fluids, they show solid-like properties for instant re- sponse and show fluid-like properties on long time scales The microstructures of the fluid influence the flow field and are modified by flow field in turn Thus, viscoelastic fluid flows are more complex than traditional Newtonian fluid flows

Modeling viscoelastic flows is important for understanding and predicting the

3 behavior of processes, and thus for designing optimal flow configurations and for se- lecting operating conditions Modeling viscoelastic flows presents many difficulties such as setting appropriate constitutive models for viscoelastic fluids, obtaining con- verged solutions numerically, reaching high flow intensity, and imposing appropriate boundary conditions

Viscoelastic liquids flowing in complex two- and three-dimensional domains are generally modeled by introducing the viscoelastic stress o' and adding an extra equa- tion to the momentum-continuity pair, usually of rate-type (Crochet, Davies and Walters 1984, Marchal and Crochet 1987, Rajagopalan, Armstrong and Brown 1990,

Baaijens 1998b, Keunings 2000, Tanner 2000, Owens and Phillips 2002), e.g., Oldroyd-

B (Oldroyd 1950, Bird Armstrong and Hassager 1987), Giesekus (Giesekus 1982),

Leonov (Leonov 1976, Leonov and Prokunin 1994), PTT (Phan-Thien and Tanner

1977, Phan-Thien 1978), FENE-P (Bird, Armstrong and Hassager 1987), FENE-CR (Chilcott and Rallison 1988), etc More recently, viscoleastic liquids have been mod- eled by introducing microstructural variables that represents the local state of the liquid—e.g., the conformation tensor M for polymer solutions (Grmela and Carreau

1987, Beris and Edwards 1994, Jongschaap, de Haas and Damen 1994)—and writing transport equations for the microstructural variables and a constitutive relationship between such microstructural variables and the viscoelastic stress (Guénette, Abdel-

malek, Fortin, Carreau and Grmela 1992, Beris and Sureshkumar 1996, Sureshkumar,

Beris and Handler 1997, Pasquali and Scriven 2002)

In this thesis, 3-D viscoelastic flows are successfully modeled with the

confor-mation tensor model This conforconfor-mation tensor depends on the flow, and in turn modifies the flow by contributing to the total stress The eigenvalues and eigenvec- tors of the conformation tensor are the local expectation values of the state of strain and stress of an ensemble of polymer molecules in the flow; thus, conformation tensor models seem to be a good compromise of computational cost and physical accuracy for modeling complex flows

So far, fully 3-D computations based on conformation tensor models have not been attempted In this thesis, the DEVSS-G/SUPG finite element method is applied to simulate steady viscoelastic flows modeled by the conformation tensor A 3-D 4:1 contraction flow is simulated and validated with 2-D results when the channel depth

is much greater than the channel width

A new, convenient way of imposing open-flow boundary conditions for the trans- port equation of the conformation tensor is developed in this thesis for both 2-D and 3-D viscoelastic flows In general, the distribution of the conformation tensor along open boundaries is unknown A general inflow boundary condition based on solving the coupled algebraic equations of fully developed flow at the inflow is proposed to circumvent this difficulty The transport equation for the conformation tensor can be written as

v:VM= Z(Vv,M) -ớ(M) (1.1)

where M is the conformation tensor, v is the velocity, V denotes the gradient in space, and Z and G are two tensor-valued functions (possibly nonlinear) which specify the model completely On rectilinear fully developed inflow boundary, v- VM = 0, and

thus equation 1.1 becomes

is also compared with the traditional boundary conditions in a flow around a cylinder The comparison results show that this developed boundary condition is better than all the other boundary conditions except the boundary condition of imposing analytical values there However, the analytical boundary condition doesn’t exist in most cases; therefore, this new method of imposing boundary condition is the best choice for most

cases

1.3 Parallelizable approximate inverse GMRES preconditioner

Solving equations arising from general computational fluid dynamics is not a challenge any more, however, equations reduced from free surface/interface flows present a big challenge in convergence, especially when the fluids of flows have elasticity Because surface and viscoelastic forces are comparable or more impor- tant than viscous ones, there are large non-diagonal contributions in the momen- tum equations that come from the deformation of the free surfaces or elastic bound- aries, and from the microstructural elastic stress Also, the boundary conditions on free surface that couples the fluid and the position produce significant non-diagonal

contributions Thus, the Jacobian matrix generated in the Newton’s iteration is ill-conditioned with large off-diagonal entries In solving 2-D free surface flows,

frontal solver (Duff, Erisman and Reid 1986), which is a direct solver, has been

widely used (de Almeida 1995, Carvalho 1996, Pasquali 2000) However, in com-

puting 3-D free surface flows, the direct solver is not practical for the large scale system, thus an iterative method, GMRES (Generalized Minimum Residual) solver (Saad and Schultz 1986), has been applied (Cairncross, Schunk, Baer, Rao and Sackinger 2000, Baer, Cairncross, Schunk, Rao and Sackinger 2000) GMRES con- verges very slowly when applied to this poorly conditioned system Therefore, an effective preconditioner is highly desired to speed GMRES convergence

The preconditioner developed here is a Sparse Approximate Inverse Preconditioner (SPAI) The preconditioner is computed explicitly by minimizing the Frobenius-norm based on the sparsity pattern of the banded Jacobian matrix as follows,

n

min ||I— AP||} = min Ề ` |le; — Apjll (1.3)

j=l where I is the identity matrix, A is the banded Jacobian chosen for computing pre- conditioner P, n is the total number of unknowns, e,; and p; are the jth column of

I and P, respectively The computation of the preconditioner involves solving a set

of uncoupled least squares problems, which can be parallelized easily on distributed memory machines The Message Passing Interface (MPI) (Dongarra, Hempel, Hey

and Walker 1993, Waler 1994) is used to parallelize the code for the CPU intensive

parts, which include (1) computation of elementary Jacobian matrices, (2) construc- tion of inverse preconditioner, and GMRES iterations Elementary Jacobian matri-